A:
Chandra took a brief look at the Earth in February last year and is
right now in the process of taking a longer observation. Please check
in with the Chandra web site (http://chandra.harvard.edu/) over the next
few months since we will likely be posting an X-ray image of the Earth
here when it becomes available.

We've checked the data archives of another X-ray telescope,
XMM-Newton, and it seems that it has not observed the Earth.
However, the small optical telescope on board the XMM-Newton took an
image of the telescope with the Earth behind it that you may find interesting:
http://spaceflightnow.com/news/0002/20xmmseesearth/

But for X-ray images of Earth, keep checking the Chandra pages in the upcoming months or sign up for our free Chandra newsletter for notification of releases